On partially integrated transport models for subgrid-scale modeling Roland Schiestel∗ IRPHE, Chˆ ateau-Gombert, 13384 Marseille, France Bruno Chaouat∗∗ ONERA , 92322 Chˆatillon, France

1

Context and issues

Mathematical turbulence modelling methods have made significant progress in the past decade for predicting various practical turbulent shear flows. Many different types of models have been developed in the past, such as RANS (Reynolds-averaged Navier-Stokes) models [1, 2]. Generally, the RANS models appear well suited to handle engineering applications involving strong effects of streamline curvature, system rotation, wall injection or adverse pressure gradient encountered for instance in aeronautics applications [3, 4, 5, 6]. Multiple-scale models [7, 8, 9] have been developed to account for spectral non-equilibrium in the framework of one-point closures. Others works [10, 11] have been made then in this framework. Important works have been also devoted to the two-point approach to extend these closures to the case of non-homogeneous turbulence. After the work of Cambon et al. [12] dealing with the extension of EDQNM (eddy-damped quasi-normal Markovian) closures, several efforts have been pursued [13]. On the other hand, as shown for instance by Lesieur [14], the LES (large eddy simulations) method using subgrid modelling techniques [15, 16] favoured by the continuous increase in computer power and speed has been extensively developed. All these various approaches have often been developed along independent lines and the connection between them is generally not clearly established. So, there is a real need to unify these points of view in a coherent manner in order to easily bridge these apparently different models [17]. In this line of thought, let us mention that recently, new turbulence models that take advantage of RANS and LES approaches based on hybrid zonal methods [18, 19, 20] or on a hybrid continuous method with “seamless coupling ” [21, 22] are now currently developed for simulating practical turbulent flows. These models are useful for calculations on relatively coarse grids when the spectral cutoff is located before the inertial zone. The hybrid continuous method [21, 22] presents major interest on a fundamental point of view because it bridges different levels of description ∗

Senior Scientist, CNRS. E-mail address: [email protected] Senior Scientist, ONERA. E-mail address: [email protected]

∗∗

1

in a consistent way [23]. With a particular emphasis upon the connection between RANS and LES, we shall show in this paper how the continuous hybrid formulation can be developed. This method is based on the spectral Fourier transform of the two-point fluctuating velocity correlation equations with an extension to nonhomogeneous turbulence [17]. In particular, the partial integration based on spectrum splitting, gives rise to PITM method (partial integrated transport models) [21, 22]. This approach can yield subfilter transport models that can be used in LES or in hybrid methods, providing some appropriate approximations are made. The method is well appropriate for calculating non-equilibrium turbulent flows. In this paper, some applications will be then considered for illustrating the potentials of this approach. In the present paper, we shall rely upon a theoretical method based on mathematical physics formalism to allow transposition of turbulence modelling from RANS to LES [17]. The recent scientific literature shows increasing interest in the use of more advanced models in subgrid-scale closures, including subfilter algebraic stress models or stress transport models inspired from RANS [24]. This can be related also to the hybrid RANS/LES approach with seamless coupling [20, 23].

2 2.1

PITM approach to subgrid-scale turbulence models General formalism

As usually made in large eddy simulations, the spectrum is then portioned using a cutoff wave number κc . In classical LES this cutoff is located in the beginning of the inertial range of eddies but in the present approach, like in very large eddy simulations, the cutoff may be located before the inertial range. For convenience, another wave number κd located at the end of the inertial range of the spectrum can also be used, assuming that the energy pertaining to higher wavenumbers is entirely negligible. This practise avoids considering infinite limits and molecular viscosity effects in the far end of the spectrum. When non-homogeneous turbulence is considered (this is the usual case), the concept of tangent homogeneous space at a point of the non-homogeneous flow field must be used. In this case, it is then possible to > define the large scale fluctuations (resolved scales) u< i and the fine scales (modelled scales) ui through the relations using the wave number κ Z < ui = ub0 i (X, κ) exp (jκξ) dκ (1) |κ|≤κc

u> i =

Z

ub0 i (X, κ) exp (jκξ) dκ

(2)

|κ|≥κc

If large eddy simulations make use of a filtering operation instead of statistical averaging, it is of interest to remark that the previous definition is indeed a filter operating in Fourier space. But it is a particular filter with interesting properties : if integration is performed in κ in the tangent homogeneous space, then, the quantity obtained becomes a function of X, and it is the usual statistical mean (see figure 1). So, the previous filter, sometimes called the statistical filter, is well suited to bridge RANS and 2

Figure 1: Sketch of tangent homogeneous space hypothesis

LES. Then, the instantaneous velocity ui can be decomposed into a statistical part hui i, a large scale > < > fluctuating u< i and a small scale fluctuating ui such that ui = hui i + ui + ui . The first two terms < correspond to the filtered velocity u¯i such that u¯i = hui i + ui . The velocity fluctuation u0i contains > a large-scale and a small-scale parts, u0i = u< i + ui . This particular filter, as a spectral truncation, presents also some additional useful properties that are not verified for progressive filters. In particular, it can be shown [8] that large scale and small scale fluctuations are uncorrelated hϕ> ψ < i = 0 implying for instance the relation

 > > < Rij = hui uj i − hui i huj i = hu0i u0i i = u< + ui uj (3) i uj The transport equation for the filtered Navier-Stokes equations takes the form ∂τ( ui , uj ) ∂ 1 ∂ p¯ ∂ 2 u¯i ∂ u¯i + (¯ ui u¯j ) = − +ν − ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj

(4)

in which, following Germano’s derivation [25], the subgrid-scale tensor is defined by the relation (τij )sgs = τ (ui , uj ) = ui uj − u¯i u¯j

(5)

The work of Germano [25] shows that the transport equation for the subgrid-scale tensor takes a generic form if it is written in terms of central moments. The resulting equation can be rearranged as  ∂τ (ui , uj ) ∂  + τ (ui , uj )¯ uk = ∂t ∂xk ∂ u¯j ∂ u¯i −τ (ui , uk ) − τ (uj , uk ) ∂xk ∂xk   ∂ui ∂uj +τ p, + ∂xj ∂xi 1 ∂τ (p, ui ) 1 ∂τ (p, uj ) ∂τ (ui , uj , uk ) − − − ρ ∂xj ρ ∂xi ∂xk   2 ∂ τ (ui , uj ) ∂ui ∂uj +ν − 2ντ , (6) ∂xk ∂xk ∂xk ∂xk 3

¯ (f, g) − f¯g¯h ¯ with the general definition τ (f, g) = f g − f¯g¯ and τ (f, g, h) = f gh − f¯τ (g, h) − g¯τ (h, f ) − hτ for any turbulent quantities f , g, h. Equation (6) will then be solved numerically in space and time. This equation is fluctuating but has a similar form as the equations used in statistical multiple scale models [17]. Using the definition D/Dt = ∂/∂t + u¯k ∂/∂xk , equation (6) reads D(τij )sgs = (Pij )sgs + (Ψij )sgs + (Jij )sgs − (ij )sgs Dt

(7)

where in this equation, the production term (Pij )sgs is (Pij )sgs = −(τik )sgs

∂ u¯j ∂ u¯i − (τjk )sgs ∂xk ∂xk

(8)

The corresponding equation for subfilter energy is obtained by the half trace Dksgs = Psgs + Jsgs − sgs Dt

(9)

where Psgs = (Pmm )sgs /2 and sgs = (mm )sgs /2. Because of the nice properties of the truncation filter in Fourier space, the mean statistical and filtered equations can both be written in a similar form. As a consequence, we shall assume that closure approximations used for the statistical partially averaged equations also prevail in the case of large eddy numerical simulations. Let us mention that the present formalism is in fact the essence of the PITM model, first developed by Schiestel and Dejoan [21] for the transport equation (9) of the subgrid-scale turbulent energy ksgs = (τmm )sgs /2 and subsequently by Chaouat and Schiestel [22] for the transport equation (7) of the subgrid-scale turbulent stress tensor (τij )sgs .

2.2

Two-equation subfilter model

For LES or hybrid RANS/LES approaches, this level of closure is composed of an equation for subfilter turbulence energy coupled with a dissipation rate equation. The transport equation of the dissipation rate used in subfilter models is somewhat different from the equation usually used in statistical models. In the tangent homogeneous space, the value of the mean velocity gradient is denoted Λij . The equation of the energy spectrum balance E(κ) can be obtained by taking the Fourier transform and mean value on spherical shells of the transport equation of the two-points velocity correlation [8, 26]: ∂E = −Λij τij + T − 2νκ2 E ∂t

(10)

Integration of the basic equation (10) over the wave number range [κc , κd ], where κc is the cutoff wave number given by the filter width and κd is the splitting wave number (see figure 2), yields at high Reynolds numbers ∂ hkgs i = hPsgs i − F (κd ) + F (κc ) −  (11) ∂t 4

with the relations

Z

κd

hksgs i =

E(κ)dκ

(12)

κc

Z

κd

hPsgs i = −Λlm

τlm (κ)dκ

(13)

κc

∂κ F (κ) = F(κ) − E(κ) ∂t Z κ Z ∞ T (κ0 )dκ0 T (κ0 )dκ0 = − F(κ) =

(14) (15)

0

κ

Z

κd

κ2 E(κ)dκ

 = 2ν

(16)

κc

F represents the spectral energy rate transferred into the wave number range [κ, +∞] by vortex stretching from the wave number range [0, κ]. Equation (9) can be derived equivalently in physical space [22] with corresponding expressions for the production, transfer and dissipation (9). Considering the cutoff wave number κc given by the filter width, the splitting wave number κd is then determined by the dimensional relation  κd − κc = ζc (17) hksgs i3/2 where ζc is a coefficient which may be dependent on the spectrum shape and on the Reynolds number. Note that this relation is identical to the relation introduced in statistical multiple scale models [9]. The net flux across the splitting wavenumber κd , due to the variations of the splitting is related to the usual spectral flux by equation (14). As a consequence we obtain F(κd ) − F (κd ) ∂κd = ∂t E(κd )

(18)

Taking into account equation (18) one can easily obtain the transport equation for the dissipation rate ∂  2 = csgs 1 (hPsgs i + F (κc )) − csgs 2 ∂t hksgs i hksgs i where csgs 1 = 3/2 and csgs 2

3 hksgs i = − 2 κd E(κd )



 F(κd ) −1 

(19)

(20)

setting κd  κc , and E(κd )  E(κc ). In the case of full statistical modelling where κc = 0, equation (17) is reduced to the equation:  κd = ζd 3/2 (21) k

5

where the coefficient ζd is a numerical constant chosen such that κd is located after the inertial range. By taking the derivative of equation (21) with respect to time, using equation (18) , another formulation of the standard dissipation rate equation is then obtained ∂  2 = c1 P − c2 ∂t k k where c1 = 3/2 and 3 k c2 = − 2 κd E(κd )



 F(κd ) −1 

(22)

(23)

This is in fact the usual  equation used in statistical closures. Equations (20) and (23) show that the coefficients csgs 1 and csgs 2 are functions of the spectrum shape. Keeping in mind that the dissipation rate  must remain the same regardless the location of the wave number κc , comparing equation (19) with equation (22) allows to express the coefficient csgs2 in a more convenient form csgs2 = c1 +

2.3

hksgs i (c2 − c1 ) k

(24)

Model calibration

The function hksgs i /k which appears in equation (24) can be calibrated by referring to the Kolmogorov law of the three-dimensional energy spectrum in the inertial wave number range in nearly equilibrium flows E(κ) = CK 2/3 κ−5/3 where CK ≈ 1.50 is the Kolmogorov constant. The subgrid-scale turbulent kinetic energy is then estimated by integrating the Kolmogorov law in the wave number range [κc , +∞[ Z ∞ 3 (25) hksgs i = E(κ)dκ = CK 2/3 κc−2/3 2 κc < Introducing a dimensionless wave number defined by ηc = κc k 3/2 /( + < ) where < = ν∂u< i ∂ui /∂xj ∂xj represents the small part of dissipation coming from the resolved scales u< i , and taking into account −2/3 equation (25), we obtain ksgs /k = 1.5 CK ηc . The statistical kinetic energy k = hksgs i + hkles i is the total turbulence energy. As it was mentioned, the total dissipation rate  + < includes now the usual part caused by the subgrid-scale fluctuating and the small part coming from the resolved scales fluctuating in order to represent the characteristic scale of the whole turbulence spectrum. As found, −2/3 the function hksgs i /k is dependent of the parameter ηc . However, this previous result is only valid in the inertial range. It is extended empirically to the general case, taking care to satisfy the limit when hksgs i tends to k, ( i.e. when ηc goes to zero). So, the coefficient csgs2 in equation (24) is modelled as follows c2 − c1 csgs2 = c1 + (26) 2/3 1 + βη ηc

where βη is a numerical constant which takes the theoretical value βη = 2/3CK ≈ 0.444 in order to −2/3 satisfy the correct asymptotic behaviour in ηc for high values ηc with the limiting conditions: 6

limηc →0 csgs2 (ηc ) = c2 , limηc →∞ csgs2 (ηc ) = c1 . In the limit of full statistical modelling, hksgs i → k and the usual RSM model is recovered while in the limit hksgs i → 0, the subgrid-scale energy is not maintained due to the fact that csgs2 → c1 and the model behaves like a DNS (but the model become useless!). The instantaneous fluctuating dissipation rate sgs verifies the relation hsgs i = . For LES, we propose a modelled transport equation for the fluctuating dissipation rate sgs , referring to equation (19). Taking into account the convective and diffusive processes as well as low Reynolds number terms for non-homogeneous flows, the fluctuating dissipation rate sgs then reads sgs (Pmm )sgs e sgs sgs Dsgs = c1 − csgs2 + (J )sgs (27) Dt ksgs 2 ksgs   νsgs  ∂sgs ∂ where ν+ (28) (J )sgs = ∂xj σ ∂xj p
2 and e sgs = sgs − 2ν ∂ ksgs /∂xn . The values of the numerical coefficients in equations (26) and (27) are c1 = 1.45, c2 = 1.9 and σ = 1.3.

2.4

Practical formulation

The two equation subfilter model is composed of the modelled equations (9) and (27) together with a 2 gradient diffusion hypothesis νsgs = cν ksgs /sgs . In a practical formulation for the case of wall bounded flows, the length scale can be computed using the normal distance to the wall L = Kx3 where K is the Von K´arm´an constant. In that condition, we use the alternative dimensionless wave number Nc = κc L instead of ηc and we introduce the modified coefficient βN in equation (26). In that framework, the 2/3 alternative functions of the subgrid-scale turbulence model are written equivalently with βη ηc = 2/3 βN Nc . The order of magnitude of the new coefficient βN is then obtained by reference to the logarithmic layer leading to the theoretical value βN ≈ 1.466. The cutoff wave number is approximated by the filter width κc = π/(∆1 ∆2 ∆3 )1/3 .

2.5

Limiting behaviour

With the tangent homogeneous space in mind, let us remark finally that when very large filter widths are used, the filter width has to be dissociated from the grid itself, because the grid must always be fine enough to capture the mean flow non-homogeneities. When the cutoff location is large then, limiting 3/2 behaviours are obtained. The length scale ksgs /sgs is equal to  3/2 3/2 k 3/2 ksgs ksgs = (29) sgs sgs k Taking into account the preceding expression of hksgs i /k, equation (29) shows that the subgrid characteristic length scale goes to the filter width 3/2 ksgs /sgs = (3CK /2)3/2 ∆/π

7

(30)


1/2 3/2 3/2 3 3/2 3/2 3 Moreover, the definition of subfilter viscosity implies νsgs = cν (ksgs /sgs ) = cν ksgs /2sgs sgs or
3/2 3 /2sgs (sgs /νsgs )1/2 . Using then the previous result on the length scale together with the νsgs = cν ksgs hypothesis of equilibrium sgs = 2νsgs hSi,j Si,j i where Sij = (∂ui /∂xj + ∂uj /∂xi )/2, one finds that the limiting behaviour for the subgrid viscosity νsgs is simply the Smagorinsky model  3 1 3CK 1/2 2 νsgs = 2 (31) c3/2 ν ∆ [2 hSij Sij i] π 2

2.6

Stress transport equation subfilter model

In the subfilter models, as usual in statistical approaches, the redistribution term (Ψij )sgs which appears in equation (7) is decomposed into a slow and a rapid part (Ψ1ij )sgs and (Ψ2ij )sgs in the subgrid-scale range. The slow term is modelled assuming that usual statistical Reynolds stress models must be recovered in the limit of vanishing cutoff wave number κc and also that the return to isotropy is increased at higher wave numbers [22], as also assumed in multiple-scale models   sgs 1 2 (Ψij )sgs = −csgs1 (32) (τij )sgs − (τmm )sgs δij ksgs 3   1 1 (Ψij )sgs = −c2 (Pij )sgs − (Pmm )sgs δij (33) 3 where csgs1 is now a continuous function of the cutoff wave number κc . The value of this coefficient can be calibrated from experiments. According to the classical physics of turbulence, the coefficient csgs1 must increase with the parameter ηc in order to increase the return to isotropy in the range of larger wave numbers. To do that, we suggest a simple empirical function csgs1 =

1 + αη ηc2 c1 1 + ηc2

(34)

where αη is a numerical constant. This function satisfies the limiting condition limηc →0 csgs1 (ηc ) = c1 . In the practical equivalent formulation, αη ηc2 = αN Nc2 . In this formulation, like in the Launder and Shima model [27], the function c1 depends on the second and third subgrid-scale invariants A2 = aij aji , A3 = aij ajk aki and the flatness coefficient parameter A = 1 − 98 (A2 − A3 ) where aij = ((τij )sgs − 32 ksgs δij )/ksgs . The term (Ψij )sgs takes into account the wall reflection effect of the pressure fluctuations and is embedded in the model for reproducing correctly the logarithmic region of the turbulent boundary layer. It is modelled according to the previous work of Gibson [28]. The diffusion process (Jij )sgs is modelled assuming a gradient law [22]   ∂ ∂(τij )sgs ksgs ∂(τij )sgs (Jij )sgs = ν + cs (τkl )sgs (35) ∂xk ∂xk sgs ∂xl where cs is a numerical coefficient which takes the value 0.22. Moreover, we assume (ij )sgs = (2/3)sgs δij . In contrast to the two-equation model, it can be mentioned that the production term (Pij )sgs is allowed 8

to become negative. In such a case, this implies that energy is transferred from the filtered motions up to the resolved motions, known as back-scatter process. So that, the PITM model for (τij )sgs and sgs is based on the modelled equations (7) and (27). However, note that equation (27) is modified for the diffusion term because of its new tensorial formulation   ∂ ∂sgs ksgs ∂sgs (J )sgs = ν + c (τjm )sgs (36) ∂xj ∂xj sgs ∂xm In equation (36), the coefficient c takes the value 0.18. For the limiting condition when the cutoff wave number goes to zero, one can see that the PITM model goes to the original model of Launder and Shima [27].

3 3.1

Some illustrative applications of PITM to LES and hybrid models Non-equilibrium turbulent flows

The PITM two-equation model (9) for the subfilter turbulent kinetic energy dissipation rate has been applied to several turbulent flows including in particular the pulsed turbulent channel flow performed by Schiestel and Dejoan [21] showing occurrence of lag effects and also the turbulent shearless mixing layer by Befeno and Schiestel corresponding to the mixing of two turbulent fields of differing scales [29]. In the present paper, we shall focus mainly on injection channel flow.

3.2

Injection induced turbulent channel flow

A channel flow with mass injection through one porous wall which undergoes the development of natural unsteadiness with a transition process from laminar to turbulent regime has been performed by Chaouat and Schiestel [22]. This case is of central interest for engineering applications in solid rocket motors (SRM). The present large eddy simulation has been made using a medium grid (400 × 44 × 80) in the streamwise, spanwise and normal directions to the wall. The velocities and turbulent stress profiles are compared with experimental data [30], and also with RSM computations obtained for the limit of the PITM model when the cutoff wave number goes to zero [31, 32]. Figure 3 shows the isosurfaces of the instantaneous spanwise filtered vorticity ω ¯ 2 = ∂ u¯3 /∂x1 − ∂ u¯1 /∂x3 in the downstream part of the channel and reveals the detail of the flow structures subjected to mass injection. The isosurfaces exhibit roll-up vortex structures in the spanwise direction, indicating the transitional and turbulent flow regime. Figure 4 shows the velocity profiles hu1 i /um normalised by the bulk velocity um in two locations of the channel at x1 = 40 cm and 57 cm. It appears that both LES and RSM computations produce velocity profiles that agree rather well with the experimental data. Figure 5 describes the streamwise turbulent stresses hu01 u01 i /um in different stations of the channel at x1 = 40 cm and 57 cm. As a result of interest, one can observe that both LES and RSM computations reasonably well predict the turbulence intensity

9

of the flow in the downstream transition location where the flow presents a turbulent regime, except, however, in the immediate vicinity of the wall region.

4

Conclusion

We have shown that the partial integration concept allows to develop subfilter turbulence transport models that can be used in LES or hybrid approaches. The concept of tangent homogeneous field, considered as deriving from the first term in the Taylor development of local mean velocity field together with the use of the spectral statistical filter are essential ingredients [17]. They allow in particular to dissociate the filter from the grid itself. Because of the filtering made in the tangent space, the method can be applied in non-homogeneous flows. As known, the total integration in the tangent space exactly produces the corresponding one-point statistical model in a consistent way. This character is important for hybrid modelling applications. On the other hand, when the filter width is small, we have shown, assuming equilibrium flows, that the proposed model is equivalent to a Smagorinsky type model (of course, provided that the mesh is finer than the filter width). The PITM concept has been considered for two-equation models (k− type models) [21] and for stress transport models [22]. Obviously, every other statistical model of the scientific literature (including two-equation models, algebraic stress models, non linear models and various stress transport models) can also be transposed in subfilter version. Some applications including flow situations with non equilibrium spectrum have been then presented for illustrating some potentials of the method. The main contribution of the present approach is therefore to bridge URANS models and LES simulations, opening a promising route of new future developments in hybrid models with seamless coupling.

References [1] G. Speziale C. Analytical methods for the development of Reynolds-stress closures in turbulence. Annual Reviews Fluid Mechanics, 23:107–157, 1991. [2] B. E. Launder. Second moment closure: Present and future ? Int. Heat and Fluid Flow, 20(4):282– 300, 1989. [3] B. Chaouat. Reynolds stress transport modeling for high-lift airfoil flows. 44(10):2390–2403, 2006.

AIAA Journal,

[4] M. A. Leschziner and D. Drikakis. Turbulence modelling and turbulent-flow computation in aeronautics. The Aeronautical Journal, 106:349–384, 2002. [5] C. L. Rumsey and T. B. Gatski. Recent turbulence model advances applied to multielement airfoil computations. Journal of Aircraft, 38(5):904–910, 2001. [6] B. Chaouat. Simulations of channel flows with effects of spanwise rotation or wall injection using a Reynolds stress model. Journal of Fluid Engineering, ASME, 123:2–10, 2001. 10

[7] K. Hanjalic, B. E. Launder, and R. Schiestel. Multiple time-scale concepts in turbulence transport modeling. In Turbulent Shear Flows n◦ 2. Ed. Springer, 1980. [8] R. Schiestel. Sur le concept d’´echelles multiples en mod´elisation des ´ecoulements turbulents. Journal de M´ecanique Th´eorique et Appliqu´e, Part I, 2, 3:417-449; Part II, 2, 4:601-628, 1983. [9] R. Schiestel. Multiple-time scale modeling of turbulent flows in one point closures. Physics of Fluids, 30(3):722–731, 1987. [10] T.T. Clark and C. Zemach. A spectral model applied to homogeneous turbulence. Physics of Fluids, 7(7):1674–1694, 1995. [11] A. Cadiou, K. Hanjalic, and K. Stawiarski. A two-scale second-moment turbulence closure based on weighted spectrum integration. Theoret. Comput. Fluid Dynamics, 18:1–26, 2004. [12] C. Cambon, D. Jeandel, and J. Mathieu. Spectral modelling of homogeneous non-isotropic turbulence. Journal of Fluid Mechanics, 104:247–262, 1981. [13] J. P. Bertoglio and D. Jeandel. A simplified spectral closure for inhomogeneous turbulence: application to the boundary layer. 5th Symposium on Turbulence Shear Flows, Ed. Springer, Corneil Univ., 1986. [14] M. Lesieur, O. M´etais, and P. Comte. Large-Eddy Simulations of Turbulence. Cambridge University Press, 2005. [15] M. Lesieur and O. Metais. New trends in large-eddy simulations of turbulence. Ann. Rev Journal of Fluid Mechanics, 28:45–82, 1996. [16] M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic subgrid-scale eddy-viscosity model. Physics of Fluids, 3(7):1760–1765, 1992. [17] B. Chaouat and R. Schiestel. From single-scale turbulence models to multiple-scale and subgridscale models by Fourier transform. To appear in Theoret. Comput. Fluid Dynamics, 2007. [18] P. R. Spalart. Strategies for turbulence modelling and simulations. International Journal of Heat and Fluid Flow, 21:252–263, 2000. [19] F. Hamba. A hybrid RANS/LES simulation of turbulent channel flow. Theoret. Comput. Fluid Dynamics, 16:387–403, 2003. [20] L. Temmerman, M. Hadziabdic, M. A. Leschziner, and K. Hanjalic. A hybrid two-layer URANSLES approach for large eddy simulation at high reynolds numbers. International Journal of Heat and Fluid Flow, 26:173–190, 2005. [21] R. Schiestel and A. Dejoan. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theoret. Comput. Fluid Dynamics, 18:443–468, 2005. 11

[22] B. Chaouat and R. Schiestel. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Physics of Fluids, 17(6), 2005. [23] K. Hanjalic, M. Hadziabdic, M. Temmerman, and M. Leschziner. Merging LES and RANS strategies: Zonal or seamless coupling ? In Direct and Large eddy Simulations 5, pages 821–831. Ed. by R. Friedrich, B. Geurts and O. M´etais, Kluwer Academic, May 2004. [24] S. Bhushan, Z. U. A. Warsi, and D. K. Walters. Modeling of energy backscatter via an algebraic subgrid-stress model. AIAA Journal, 44(4):837–847, 2006. [25] M. Germano. Turbulence: the filtering approach. Journal of Fluid Mechanics, 238:325–336, 1992. [26] J. O. Hinze. Turbulence. Mc. Graw-Hill, 1975. [27] B. E. Launder and N. Shima. Second moment closure for the near wall sublayer: Development and application. AIAA Journal, 27(10):1319–1325, 1989. [28] M. M. Gibson and B. E. Launder. Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics, 86:491–511, 1978. [29] I. Befeno and R. Schiestel. Non-equilibrium mixing of turbulence scale using a continuous hybrid RANS/LES approach. to appear in Flow, Turbulence and Combustion, 2007. [30] G. Avalon, G. Casalis, and J. Griffond. Flow instabilities and acoustic resonance of channels with wall injection. AIAA Paper 98–3218, July 1998. [31] B. Chaouat. Numerical predictions of channel flows with fluid injection using Reynolds stress model. Journal of Propulsion and Power, 18(2):295–303, 2002. [32] B. Chaouat and R. Schiestel. Reynolds stress transport modelling for steady and unsteady channel flows with wall injection. Journal of Turbulence, 3:1–15, 2002.

12

Ln E F (κ c )

F (κ d )

k − k sgs

k sgs κ κc

κd

Figure 2: Sketch of spectrum splitting

Figure 3: Isosurfaces of instantaneous filtered vorticity vector ω ¯ i = ijk ∂ u¯k /∂xj in the spanwise direction (i=2) |¯ ω2 | = 3000 (1/s). LES simulation [22]. Experimental cold flow setup of Avalon [30].

13

0.8

0.8

0.6

0.6

X3/δ

1

X3/δ

1

0.4

0.4

0.2

0.2

0

0 0

0.5

1

1.5

2

0

0.5

1

/um

1.5

2

/um

(a)

(b)

Figure 4: mean velocity profiles normalised by the bulk velocity hu1 i /um in different cross sections (a) x1 = 40 cm; (b) 57 cm; —: RSM computation [31]; - - -: LES simulation [22]; ∆: experimental data [30].

0.8

0.8

0.6

0.6

X3/δ

1

X3/δ

1

0.4

0.4

0.2

0.2

0

0 0

0.1

0.2

0.3

0

0.1

rms u1/um

rms u1/um

(a)

(b)

0.2

0.3

Figure 5: Streamwise turbulent stresses hu01 u01 i1/2 /um in different cross sections (a) x1 = 40 cm; (b) 57 cm. —-: RSM computation [31]; - - -: LES simulation [22]; ∆: experimental data [30].

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